Abstract
We consider the dynamics of rapid propagation of information (RPI) in mobile phone networks. We propose a heuristic method for identification of sequences of calls that supposedly propagate the same information and apply it to largescale realworld data. We show that some of the information propagation events identified by the proposed method can explain the physical colocation of subscribers. We further show that features of subscriber’s behavior in these events can be used for efficient churn prediction. To the best of our knowledge, our method for churn prediction is the first method that relies on dynamic, rather than static, social behavior. Finally, we introduce two generative models that address different aspects of RPI. One model describes the emergence of sequences of calls that lead to RPI. The other model describes the emergence of different topologies of paths in which the information propagates from one subscriber to another. We report high correspondence between certain features observed in the data and these models.
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Notes
 1.
As a reference point for this threshold, one can consider the following statistics provided in NielsenWire (2008). The average number of monthly calls made by a subscriber in USA is only 204.
 2.
In RPIs that contain less that five subscribers, we require dissemination leader to propagate information either to all or to allbutone user in this RPI.
 3.
This specific selection of T is justified in Sect. 7.3.
 4.
The precise values of lift of churn predictors are usually considered to be proprietary information and are not mode public. However, working with a large number of telecom companies around the world, our understanding is that this lift value matches the stateoftheart for such models.
 5.
The maximal possible value of the error is 0.5. Error value smaller than 0.1 is considered to reflect a distinctive feature.
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Acknowledgments
Preliminary version of this research was published in Dyagilev et al. (2010).
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The author was also with IBM Haifa Research Lab, Haifa, Israel during a part of this research.
Appendices
Appendix 1: Analysis of information flow tree model
We proceed to investigate the properties of some of these topologies. Assume that M ≥ 5 and consider a tree generated by the information flow tree model with parameters as above. Let N denote the total number of nodes in the tree and let E _{ i } for i = 1, 2,…,4 denote the event that the generated tree is of Topology i. The following proposition assesses one of the basic properties of these topologies, namely, the distribution \({\mathbb{P}\{N=\cdot{E_1}\}}\) of sizes of RPIs in each topology.
Proposition 1
The size distribution of RPIs of Topologies 1–4 is given by the following expressions.

(1)
For Topology 1:
$$ {\mathbb{P}}\{N=n{E_1}\}= p_0(n1)\left(p_{1,4}(0)\right)^{n1}/{\mathbb{P}}\{E_{1}\} {\mathbb{I}}_{\{n\geq M\}}, $$where
$$ {\mathbb{P}}\{E_{1}\} = \sum_{r=M1}^{\infty} p_0(r)\left(p_{1,4}(0)\right)^r $$ 
(2)
For Topology 2:
$$ {\mathbb{P}}\{N=n E_{2}\} = p_0(1)p_{1,1}(n2)\left(p_{2,1}(0)\right)^{n2}/{\mathbb{P}}\{E_{2}\}, $$where
$$ {\mathbb{P}}\{E_{2}\} = \sum_{d=M2}^{\infty} p_0(1)p_{1,1}(d)\left(p_{2,1}(0)\right)^d. $$ 
(3)
For Topology 3:
$$ \begin{aligned} {\mathbb{P}}\{N=nE_3\} &= p_0(n2)\left[(n2) p_{1,4}(1)(p_{1,4}(0))^{n3}\right]\\ &\quad\times p_{2,4}(0)/{\mathbb{P}}\{E_3\}, \end{aligned} $$where
$$ {\mathbb{P}}\{E_3\} = \sum_{d=M2}^{\infty} p_0(d)\left[d p_{1,4}(1)(p_{1,4}(0))^{d1}\right]p_{2,4}(0). $$ 
(4)
For Topology 4:
$$ {\mathbb{P}}\{N=nE_4\} = (p_{2,1}(1))^{nM}(1p_{2,1}(1)). $$
Proof
These results can be easily shown by straightforward calculations.\(\hfill{\square}\)
Appendix 2: Stability of dissemination leaders
In this section, we investigate the stability of dissemination leaders over different days. We consider a training period of three consecutive days and a testing period of 14 following days. Overall, there are 19 pairs of training and testing periods in DS1 and eight pairs in DS2.
We compare the set of dissemination leaders found in the training period to the set of dissemination leaders in the testing period. In particular, we gather the following statistics: (S1) fraction of dissemination leaders in the testing period that were also dissemination leaders in the training period; (S2) fraction of dissemination leaders in the training period that were also dissemination leaders in the testing period. (S3) fraction of RPIs in the testing period in which the dominant node was a dissemination leader in the training period. We note that statistics S1 and S3 are different since a subscriber can be a dominant node in more that one RPI during the testing period.
As a baseline for these statistics, we use synthetic data in which probability of a user becoming a dominant node in some RPI in the current day does not depend on whether it was a dissemination leader in previous day. This synthetic data can be generated by the following baseline model. We assume that there exists a general pool of L dissemination leaders and numbers n _{ i }, for i = 1,…,D, of RPIs on ith day that have a dominant node. The identities of dominant node in n _{ i } RPIs of day i are selected uniformly at random from the pool of size L (with repetitions).
To make a fair comparison of statistics in the real and synthetic data, we estimate the parameters of synthetic data model in the following straightforward fashion. We let D be the number of days in the corresponding realworld data set, hence D = 35 for the data set DS1 and D = 24 for the data set DS2. We estimate L as the total number of subscribers that were dissemination leaders on any out of D days in the data set. Finally, we let numbers n _{ i } be the actual numbers of RPIs with dissemination leader in each of D days. We denote the synthetic data that correspond to data sets DS1 and DS2 by Syn1 and Syn2, respectively. The values of statistics S1–S3 are presented in Table 5. We represent each entry by its mean value over all possible selections of training and testing period. The confidence intervals on values are chosen to be one standard deviation.
Using the Wilcoxon signedrank test [e.g., see Shao (2003)], we determine that the statistics for the real data and the corresponding synthetic data differ in a statistically significant way (with p value of 0).
We next focused on stability of dissemination leaders that appeared in more than one RPI in the training set. Specifically, we considered a method that predicts that all users that were dominant nodes in at least K = 1, 2,… RPIs in the training set will also be a dominant node in at least one RPI in the testing set. The precision–recall curve of this method [see, e.g., BaezaYates and RibeiroNeto (1999) for definition] is presented in Fig. 17. We note that in realworld data, a user that was a dominant node in a large number of RPIs during the training period has a higher chance to be a dissemination leader in the testing period. In contrast, in synthetic data the number of RPIs in which the user was a dominant node during the training period has no effect of his chances to be a dissemination leader of in the testing period.
Appendix 3: Statistical analysis of features used by the RPICP algorithm
In this section, we present statistical analysis of the features used by the RPICP algorithm. We begin by presenting the complete list of features in Table 6. In Table 7, we list the three basic statistics for each feature and each data set. First, we present the average value and the standard deviation of the feature over the set of churners, i.e., subscribers that churned during the testing period. Second, we present the same values for nonchurners, i.e., subscribers that did not churn during the testing period. It can be seen that the interval of one standard deviation around the mean overlaps significantly for churners and nonchurners, for most features. Hence, it is hard to distinguish between a churner and a nonchurner using a single feature. To make this observation precise, for each feature we calculate the Neyman–Pearson error of a classifier churner/nonchurner that relies on this single feature. Namely, we calculate the minimal possible classification error one can obtain using this single feature. As expected, the Neyman–Pearson error is high^{Footnote 5} for all of the features in both data sets. Thus, multiple features are required to classify churners, each contributing a relatively small amount of information.
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Dyagilev, K., Mannor, S. & YomTov, E. On information propagation in mobile call networks. Soc. Netw. Anal. Min. 3, 521–541 (2013). https://doi.org/10.1007/s1327801301005
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Keywords
 Mobile call network
 Dynamic behavior of networks
 Churn prediction